For the time being, the cheapest way to get to Livorno from Pisa Airport, and vice versa, is by using local transport:
By bus and train - there is currently work underway at Pisa airport to provide a continuous "Pisa Mover" train connection with Pisa Central railway station. This work is due to be completed in December 2015.
At the moment there is a bus connection from the airport to Pisa Centrale station (the Bus Pisa Mover). This runs every 10 minutes from 6am to midnight daily and takes approximately 8 minutes. It costs €1.30 each way. From Pisa Centrale there are regular direct trains to Livorno (€2.50 each way on Regional trains). Please see the Trenitalia website for the full timetable.
For updates about the new train service between Pisa airport and the station, click here
For information about competitive car hire from Pisa Airport, click here
When all else fails, a taxi from the airport to Livorno costs around 50 euros, depending on time of day and amount of luggage.
Alternatively, to book a private transfer from Pisa Airport, see the recommended companies on this page